SOUNDS OF METEOROLOGICAL ORIGIN 429

A combination of two notes having different amplitudes and different initial phases gives equations substantially the same as the above, though, of course, slightly more complicated. Similarly, a multitude of sounds merge into a quasinote whose pitch is the approximate average of those of the many constituents. Hence, the whisper of a tree, whatever its volume, has substantially the same pitch as that of its individual twigs or needles; just as the hum of a swarm of bees is pitched to that of the average bee, and the concert of a million mosquitoes is only the megaphoned whine of the type.

The final law of sound, essential to the explanation of the whispering of trees, namely, the intensity of a blend in terms of that of its constituents, has been found, by Lord Rayleigh,1 in substantially the following manner:

Let the number of individual sounds be n, all of unit amplitude and same pitch, but arbitrary phase — conditions that approach the seolian blend of a tree. Clearly, if all the individual sounds had the same phase, and unit amplitude, at any given point, their combined intensity at that point would be n2. If, however, half had one phase, and half the exactly opposite phase, the intensity would be zero. Consider, then, the average intensity when all the vibrations are confined to two exactly opposite phases, + and — . Now, the chance that all the n vibrations will have the same phase, +, say, is (H)w and the expectation of intensity corresponding to this condition (^)nn2. Similarly, when one of the vibrations has the negative phase and the n — 1 others the positive phase, the expectation is (^)nn(n — 2)2; and the whole or actual expectation

l.n* + n(n - 2)* + n(n l\n - 4)2 +• - - - (A)

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The sum of the n + 1 terms of this series is n, as may be indicated by a few numerical substitutions; or proved, as Lord Rayleigh (loc. cit.) has shown, by expanding the expression

(ex + e~x}n into the two equivalent series

2'/l + *z2 +• • • - V (Maclaurin)

and

enx _|_ we<n-2)* _|_ . . . . (Binomial)

developing the exponentials into series of algebraic terms, and, finally, assembling and equating the coefficients of #2 in the two equivalent series, and solving for n. The value of n thus found is identical with the expression (A).

1 "Encyclopedia Britannica," 9th Ed., Wave Theory; Scientific Papers, 3; 52.